Convexity theory for term structure equation:an extension to the jump-diffusion case

U.U.D.M. Project Report 2010:5

Examensarbete i matematik, 30 hp

Handledare och examinator: Johan Tysk

Maj 2010

Convexity theory for term structure equation:

an extension to the jump-diffusion case

Kailin Zeng

Contents

Abstract

1

Acknowledgement

2

1 Introduction

...

3

2 Introduction to Lévy process

...

4

3 The financial model and definitions

...

7

4 Necessary conditions for preserving of convexity

………

9

5 Regularity for the value function and the parameters

...

12

6 Sufficient condition for preserving of convexity

...

13

7 Monotonicity

of

the parameters

………...

17

Convexity theory for term structure equation:

an extension to the jump-diffusion case

Abstract

This paper addresses the convexity theory for term structure equation, when the short rate is modeled under a controlled jump diffusion process. Firstly, we give the definition of locally convexity preserving and PIDE operators associated with continuous and jump fluctuations. After this, the necessary condition is derived through examining the locally convexity preserving conditions for the corresponding operators. To derive the sufficient condition for convexity of the model, we construct an auxiliary function, which is supersolution in a convexity sense. With suitable regulations on model parameters and restrained drift and jump size, the convexity of the model is guaranteed. Lastly, it follows that the value function is increasing in the volatility, jump size and jump intensity, while decreasing in the drift of the short rate.

Acknowledgements

This thesis would not be possible without assistance and support from many sides. A special thanks is due to my thesis mentor Professor Johan Tysk, who not only helped work out an outline but gave me the insight to construct the main idea of this thesis. His patience, inspiration and guidance never ceased and kept me continuously focusing on further reading and thinking, even though I changed the subject of my thesis halfway through.

I am also indebted to Erik Ekström, Svante Janson, Huyên Pham for their prominent research on convexity theory and jump diffusion processes, which pushed me to write on this topic; to the teachers in the mathematical finance program who gave me fundamental knowledge in mathematics and financial economics; to Julie Loucks for her special contributions.

Needless to say, I am grateful to my love. Her affection and support created a very harmonistic environment for my research.

1 Introduction

A model for the short rate is convexity preserving, if the price of any European claim is convex as a function of the underlying short rate at all times prior to maturity. Our main motivation for studying the convexity property is attributed to the connection between convexity preserving models and the monotonic property with respect to various parameters of the models. The parameters include drift in time, the diffusion coefficient recognized as volatility, jump intensity of the underlying Poisson process and the amplitude of each jump vibration. The convexity theory for the term structure equation in the diffusion case has been thoroughly studied in Ekström and Tysk [3]. It is shown that when the jump component is added to the original diffusion model, a new auxilary function and adjustive regulations of coefficients are needed.

Firstly, according to Ekström and Tysk [1], a necessary condition for the convexity preserving jump-diffusion short rate model can be derived. For convenience we use the same notation as in Ekström and Tysk [1]:

U

_τ

= Α + U

U

B

(1.1) The price is the unique viscosity solution to the parabolic integro-differential equation in (1.1), where

A

is associated with continuous fluctuations of the model and B is the corresponding jump part of the model. We denote

M

= +

A

B

for further usage. In this paper, we consider the one-dimensional model, which is the simple case given in Ekström and Tysk [1].

Already known in Ekström and Tysk [1], locally convexity preserving (LCP)

M

presents as a necessary condition for models preserved with convexity. In addition,

M

is LCP if and only if both

A

and

B

are LCP. This fact still holds even if the operator A is slightly different. The sketch of proof is given in section 4. Given the exact integro-differential equation for jump-diffusion term structure model, we can find the necessary condition for the convexity preserving property. This is done through looking at the corresponding LCP conditions which must hold for both

A

and

B

in the model. More details will be presented in section 4.

Another question this paper addresses is how to find the sufficient condition for the convexity preserving jump-diffusion term structure model. Figuring out the sufficient conditions seems more complicated than it did previously. However, thanks to the analytical methods in Ekström and

Tysk [2] and Ekström and Tysk [3], it is still possible to answer this question. These articles suggest that we first study the conditions, like LCP for instance, to secure convexity for the model, and then try to find conditions to fulfill what we presume (LCP). Section 6 is served for concrete implementation steps.

Section 7 serves as an additional section for the study of monotonicity in the model parameters. This work has been done in Ekström and Tysk [2] for options priced in models with jumps, as well as in Ekström and Tysk [3] for diffusion term structure models. Analogical consequences are derived in the jump-diffusion term structure models.

2 Introduction to Lévy process

Definition 2.1 Compound Poisson process A compound Poisson process with intensity

0

λ

>

and jump size distribution

f

is a stochastic process

X

_t defined as

1 t N t i

X

=

=

∑

Y

i (2.1) where jump sizes

Y

_iare i.i.d. with distribution

f

and ( ) is a Poisson process with intensity

t

N

λ

, independent from

( )

Y

_{i i≥1}.

A compound Poisson process can be decomposed as a superposition of independent Poisson processes with different jump sizes. The sample paths of compound Poisson process are cadlag1 piecewise constant functions. Moreover, from Proposition 3.3 in Cont and Tankov [4], we know a process called a compound Poisson process if and only if it is a Lévy process and its sample paths are piecewise constant functions. The Lévy process is intensely studied since it has been able to represent some empirical characteristics in derivatives pricing.

Definition 2.2 Poisson random measure A Poisson random measure is a random measure on to describe the jumps of a compound Poisson process : for any measurable

set :

[0, )

d

×

∞

0

(

X

_{t t}

)

_≥

[0, )

d

B

⊂

×

∞

( )

#{( ,

)

}

X t t

J

B

=

t X

−

X

₋

∈

B

(2.2) 1

For every measurable set

B

⊂

d,

J

_X

([ , ]

t t

_{1 2}

×

B

)

counts the number of jumps of

X

between and such that their jump sizes are in

1

t

t

₂

B

.

Definition 2.3 Lévy process A cadlag stochastic process

(

X

_{t t}

)

_≥0on

(

Ω

, ,

F

Ρ

)

with values in such that is called a Lévy process if it possesses the following properties:

d

0

0

X

=

1. Independent increments: for every increasing sequence of times , the random variables 0

...

n

t

t

0

,

1 0

,...,

n tn1 t t t t

X

X

X

X

X

−

−

−

are independent.

X

2. Stationary increments: the law of _{t h+}

−

X

_t does not depend on t. 3. Stochastic continuity:

∀ >

ε

0

, 0

lim (

_{t h t}

)

0

h

P X

₊

X

ε

→

−

≥

=

.

Definition 2.4 Lévy measure Let

(

X

_{t t}

)

_≥0 be a Lévy process on d. The measure

υ

on defined by:

d

( )

A

E

[#{(

t

[0,1] :

X

_t

0,

X

_t

A

}]

υ

=

∈

Δ

≠ Δ ∈

(2.3) is called the Lévy measure of

X

:

υ

( )

A

is the expected number, per unit time, of jumps whose size belongs to

A

.

We also define the intensity measure of a Poisson random measure on as

X

J

d

×

[0, )

∞

(

dx dt

)

(

dx dt

)

μ

×

=

υ

. Furthermore, by proposition 3.5 in Cont and Tankov [4], we can express the intensity measure

μ

(

dx dt

×

)

=

υ

(

dx dt

)

=

λ

f dx dt

(

)

.

Finally, we define the compensated Poisson random measure

J

_X by subtracting from its intensity measure: X

J

( )

( )

( )

X X

J

B

=

J

B

−

μ

B

with

B

⊂

d

×

[0, )

∞

.

Proposition 2.1 Lévy-Itô decomposition Let

(

X

t t

)

≥₀be a Lévy process on and

d

υ

its Lévy measure, given by Definition 2.4. 1.

υ

is a Radon measure on d

\ {0}

and verifies:

2 1

(

)

x≤

x

υ

dx

< ∞

∫

1

(

)

x≥

υ

dx

< ∞

∫

2. The jump measure of

X

, denoted by

J

_x, is a Poisson random measure on with intensity measure

[0, )

d

_×

_∞

(

dx dt

)

(

dx dt

)

μ

×

=

υ

.

3. There exist a vector

γ

and a d-dimensional Brownian motion with covariance matrix A such that:

0

0

lim

l t t t t

X

t

B

X

X

ε ε

γ

→

=

+

+

+

, where (2.4) 1, [0, ]

(

)

l t X x s t

X

xJ

ds dx

≥ ∈

=

∫

×

X

1, [0, ]

{

(

)

(

) }

t X x s t

x J

ds dx

dx ds

ε ε

υ

≤ ≤ ∈

=

∫

×

−

1, [0, ]

(

)

X x s t

xJ

ds dx

ε≤ ≤ ∈

=

∫

×

The terms in (2.4) are independent and the convergence in the last term is almost sure and uniform in on

t

[0

, ]

T

.

Proposition 2.2 Itô formula for scalar Lévy processes

Let

(

X

_{t t}

)

_≥0 be a Lévy process and

f

:

→

a

C

2 function, then

2 0 _{0 0} 2 ₀

( ,

_t

)

(0,

)

t _s

( ,

_s

)

t _xx

( ,

_s

)

t _x

( ,

_s

)

_s

f t X

=

f

X

+

∫

f s X ds

+

∫

σ

f

s X ds

+

∫

f s X dX

(2.5) 0 , 0

[ ( ,

)

( ,

)

( ,

)]

s s s s s x s s t X

f s X

₋

X

f s X

₋

X f s X

≤ ≤ Δ ≠

+

∑

+ Δ

−

− Δ

₋

( ,

)

( ,

)

,

t N t t i

For the proof schedule, one can read, for instance, chapter 8 in Cont, Tankov [4]. Here, we give the explicit presentation of formula (2.4) for the jump-diffusion case.

For the one dimensional jump-diffusion process, defined by the sum of a drift term, a Brownian stochastic integral and a compound Poisson process can be expressed as:

0 _{0 0} 1 t s s s i

X

γ

s X ds

σ

s X dW

X

=

+

+

+

∑

Δ

=

∫

∫

(2.6)

X

where

γ

and

σ

are continuous nonanticipating2 processes with

2 0

[

T

( ,

_t

) ]

E

∫

σ

t X dt

< ∞

Where the last term of the right side of (2.5) can be expressed by the Poisson random measure:

1 [0, ]

(

i X i t

)

t N

X

xJ

ds dx

= ×

Δ

=

×

∑

_∫

(2.7) 2

A stochastic process is said to be nonanticipating with respect to the information structure or

[0, ]

(Xt t)∈ T

[0, ]

(Ft t)∈ T Ft-adapted if, for eacht∈[0, ]T , the value of Xtis revealed at timet: the random variable t

Plugging (2.6) and (2.7) into (2.5), and we calculate each term: 0

(

)

0

( ,

) ( ,

)

0

( ,

) ( ,

)

t t t x s s x s s x s s s

f

X dX

=

f s X

γ

s X ds

+

f s X

σ

s X dW

∫

∫

∫

[0, ]

( ,

)

(

)

x s X t

xf s X

J

ds dx

×

+

_∫

×

− (2.8) And 0 , 0

[ ( ,

)

( ,

)

( ,

)]

s s s s s x s s t X

f s X

₋

X

f s X

₋

X f s X

≤ ≤ Δ ≠

+ Δ

−

− Δ

∑

[0, ]

[ ( ,

_s

)

( ,

_s

)

_x

( ,

_s

)]

_X

(

)

t

f s X

₋

x

f s X

₋

xf s X

₋

J

ds dx

×

=

∫

+ −

−

×

(2.9)

Rearranging all these terms, we get the final Itô formula representation for Jump-diffusion process: 2 0 _{0 0} 2 ₀

( ,

_t

)

(0,

)

t _s

( ,

_s

)

t _xx

( ,

_s

)

t _x

( ,

_s

) ( ,

_s

)

f t X

=

f

X

+

∫

f s X ds

+

∫

σ

f

s X ds

+

∫

f s X

γ

s X ds

0 [0, ]

( ,

) ( ,

)

[ ( ,

)

( ,

)]

(

)

t x s s s s s X t

f s X

σ

s X dW

f s X

₋

x

f s X

₋

J

ds dx

×

+

∫

+

∫

+ −

×

(2.10)

3 The financial model and definitions

Assume the short rate model modeled by a jump-diffusion process

X t

( )

satisfying

the stochastic differential equation:

( )

( (

), )

( (

), )

( )

_X

( ,

)

dX t

α

X t

t dt

β

X t

t dW t

yJ

dt dy

+∞ −∞

=

−

+

−

+

_∫

(3.1)

Here, the short rate is interpreted by drift term, diffusion term and a compensated

Poisson random measure term which counts the jumps of size

y

over time interval

. Also, we denote

[0, ]

T

λ

( )

t

the jump intensity. For the reason we can better use the consequences in Ekström and Tysk [1] and [2], it is intuitive to use a uniform distributed label

to interpret the jump size at time with

[0,1]

z

∈

t

φ

(

X t

(

−

), , )

t z

. Then, the intensity measure has been changed into the form

μ

( ,

dt dz

)

=

υ

(

dz dt

)

=

λ

( ) (

t f dz dt

)

=

λ

( )

t dzdt

. The Poisson random measure now counts the jumps on

[0, ] [0,1]

T

×

. Hence the short rate can be expressed as below:

1

0

( )

( (

), )

( (

), )

( )

( (

), , )

_X

( ,

)

dX t

=

α

X t

−

t dt

+

β

X t

−

t dW t

+

∫

φ

X t

−

t z J

dt dz

(3.1) Remark:

J

X

( ,

dt dz

)

=

(

J

X

−

μ

)( ,

dt dz

)

is defined as compensated Poisson random measure,

where is called Poisson random measure, counting the jump times in within jump size . Besides,

( ,

)

X

J

dt dz

dt

dz

μ

( ,

dt dz

)

=

λ

( )

t dtdz

is the corresponding intensity measure. Here, we assume that

W t

( )

and

J

X

( ,

dt dz

)

are independent of each other.

So far, the Itô formula for our short rate model (3.1) can be derived using proposition 2.2. The corresponding term [0, ]

( ,

)

(

)

x s X t

xf

s X

J

ds dx

×

×

∫

in (2.8) has been changed in our model:

[0, ] [0,1]

( (

), , )

_x

( ,

_s

)

_X

(

)

t

X s

s z f

s X

J

ds dz

φ

×

−

×

∫

Equation (2.9) can be expressed in slightly different way:

0 , 0

[ ( ,

)

( ,

)

( ,

)]

s s s s s x s s t X

f s X

₋

X

f s X

₋

X f s X

₋ ≤ ≤ Δ ≠

+ Δ

−

− Δ

∑

[0, ] [0,1] [ ( , _s ( ( ), , )) ( , _s ) ( ( ), , ) _x( , _s )]( _X( ) ( )) t f s X ₋ φ X s s z f s X ₋ φ X s s z f s X ₋ J ds dz ds dz × =

_∫

+ − − − − × + υ

Rearranging these terms to obtain the corresponding Itô decomposition:

2 0 _{0 0} 2 ₀

( ,

_t

)

(0,

)

t _s

( ,

_s

)

t _xx

( ,

_s

)

t _x

( ,

_s

) ( ,

_s

)

f t X

=

f

X

+

∫

f s X ds

+

∫

β

f

s X ds

+

∫

f s X

α

s X ds

0 [0, ] [0,1]

( ,

) ( ,

)

[ ( ,

( (

), ))

( ,

)]

(

)

t x s s s s s t

f s X

s X dW

X

f s X

X s

z

f s X

J

ds dz

β

φ

− − ×

+

+

+

−

−

∫

∫

×

s [0, ] [0,1]

[ ( ,

_s

( (

), ))

( ,

_s

)

( (

), , )

_x

( ,

)]

(

)

t

f s X

₋

φ

X s

z

f s X

₋

φ

X s

s z f s X

ds

dz

×

+

∫

+

−

−

−

−

₋

υ

Let : , then using proposition 3.16 in Cont and Tankov [4], we obtain the infinitesimal generator of

(

t

)

U X

×

[0, ]

T

→

t

2

1

( ( ), )

(

)

( (

), )

(

)

2

x t xx t

LU

=

α

X t t U

X

+

β

X t

−

t U

X

1 0

( ( )

( ( ), , ))

( ( ))

( ( ), , )

_x

(

_t

) (

)

U X t

φ

X t t z

U X t

φ

X t t z U

X

dz

+

_∫

+

−

−

υ

(3.2)

When we introduce the random jumps into the original diffusion model, the completeness of the model does not hold anymore. According to the second fundamental theory of mathematical finance, the equivalent martingale measures will not be unique. In this paper, we choose the risk neutral measure under which all discounted prices of contingent claims are martingales. Hence, we attain the partial integro-differential equation (PIDE) for function U:

×

[0, ]

T

→

+

(3.3)

( , 0)

( )

U

LU

xU

U x

g x

τ

=

−

⎧

⎨

₌

⎩

Where

τ

is the time after a standard change

τ

= −

T

t

and

g

is a continuous pay-off function. Let 2

1

( , )

( , )

2

x xx

AU

=

α

x t U

+

β

x t U

−

x

1 0

(

( , , ))

( , )

(

U

, , )

_x

(

)

BU

=

∫

U x

+

φ

x t z

−

U x t

−

φ

x t z U

υ

dz

(3.4) Then, we have found equation (1.1) in our special case with and stated above. It is ready to study the first problem introduced in the beginning.

AU

BU

4 Necessary conditions for preservation of convexity

Following theorem 4.2 Ekström and Tysk [1], convexity preserving models are locally convexity

preserving (LCP) models for sure, therefore we firstly introduce the notion of LCP models.

Definition 4.1 Locally convexity preserving (LCP) An operator

M

defined above is LCP at a point

(

x t

₀

, )

₀

∈ ×

[0, ]

T

if 2 0 0

(

)(

, )

x

Mf

x t

0

∂

≥

is tenable for all convex functions

f

∈

C

4

(

+

)

∩

C

2_pol

(

+

)

with

f

_xx

(

x

₀

)

=

0

. The operator

M

is called LCP if it is LCP at all points

( , )

x t

∈ ×

[0, ]

T

.

Fortunately, theorem 4.2 in Ekström and Tysk [1] still holds even though the operator

A

has additive terms as stated in (3.4). The model is convexity preserving only if the operator

M

is LCP at all points

( , )

x

τ

∈ ×

[0, ]

T

. Then for any

f

∈

C

_α4

(

×

[0, ])

T

∩

C

_pol2

(

×

[0, ]

T

)

, whether the model defined in section 3 stays convexity preserving only if its two operators

A

and

B

are both LCP at all points

( , )

x

τ

∈ ×

[0, ]

T

?

Firstly, we examine if the lemma 4.3 in Ekström and Tysk [1] still holds under model in section 3. The auxiliary functions

ϕ

and

ψ

are easily extend to functions:

(

−∞ +∞ →

,

)

[0,

+∞

)

. Let the function

h

δto be:

0 0 | |/ 1 2 0 0 _{| |/}

( (

(

))(

))

x x

( )

(

)

x x

h

δ

f

x

x

x

x

δ

s ds

f x

δ

ϕ δ

−

δ

−

ψ

− −

=

−

−

+

_∫

−

₀

Then follow the schedule in lemma 4.3 we derive the consequence that if

M

is LCP, then also

B

is LCP.

In addition, lemma 4.4 in Ekström and Tysk [1] is applicable to our model under the corresponding auxiliary functions suitable extended in

(

−∞ +∞

,

)

.

As a result, we obtain that if the model in section 3 is convexity preserving, then the operators

A

and

B

defined above are both LCP at all points

( , )

x

τ

∈ ×

[0, ]

T

.

Theorem 4.1 The model in section 3 is convexity preserving only if the following conditions are fulfilled: 2

(

2)

2

β

U

xxxx

α

xx

U

x

1

0

+

−

≥

and 1

0

≥

)

2 0

((1

+

φ

x

)

U

xx

(

x

+

φ φ

)

+

xx

U

x

(

x

+

φ φ

)

−

xx

U

x

( ))

x dz

∫

for all convexity functions

U

∈

C

_α4

(

×

[0, ])

T

∩

C

2_pol

(

×

[0, ]

T

with

U

_xx

( , )

x

τ

=

0

at all points

( , )

x

τ

∈ ×

[0, ]

T

.

Proof: According to previous discussion, it is equal to find corresponding necessary conditions for both operator

A

and

B

. Following the calculations in Ekström and Tysk [2] and Ekström and Tysk [3] with respect to equation (3.3) and letting 2

2

1

β

=

β

, we have 2 2

(

)

(

x

MU

x

AU

B

)

∂

= ∂

+ U

(2

)

(

2

)

(

2)

xxxx x xxx xx x xx xx x

U

U

x U

β

β

α

β

α

α

=

+

+

+

+

−

+

− U

)

1 ₂ 0

((1

x

)

U

xx

(

x

)

xx

U

x

(

x

)

U

xxx

(

x

λ

φ

φ φ

φ φ

+

∫

+

+

+

+

−

(1 2

φ

_x

)

U

_xx

( )

x

φ

_xx

U

_x

( ))

x dz

− +

−

(4.1） By the definition of LCP, Here

U

is convex with

U

_xx

( , )

x

τ

=

0

at some point

( , )

x

τ

. Then

( , )

xx

U

x

τ

has local minimum at point

( , )

x

τ

, therefore we attain at point

0

xxxx xxx xx

U

≥

U

=

U

=

( , )

x

τ

.

Then we obtain the necessary conditions for both

A

and

B

:

1 ₂ 0

(

2)

0

((1

)

(

)

(

)

( ))

xxxx xx x x xx xx x xx x

U

U

U

x

U

x

U

x dz

β

α

φ

φ φ

φ φ

⎧

+

−

≥

⎪

⎨

0

+

+ +

+ −

≥

⎪⎩

∫

(4.2)

Following theorem 5.3 in Ekström and Tysk [3], if the payoff function is convex and decreasing, then the necessary convexity preserving condition for operator

g

A

is

α

_xx

− ≤

2

0

. Similarly, in the condition that payoff function is convex, the necessary convexity preserving condition for operation

g

B

is

φ

_xx

( , , ) ( , , )

x t z

φ

x t z

≥

0

.

Remark: we assume

U

_x

≤

0

and

φ

≠

0

, then the following equation is sufficient for LCP :

2

0

( , , ) ( , , )

0

xx xx

x t z

x t z

α

φ

φ

− ≤

⎧

⎨

_≥

⎩

(4.3) Since U is assumed to be convex, we have consequences in (4.3) by following the fact:

(

)

( )

0

x x

U

x

φ

U

x

φ

+ −

≥

.

Given that is convex and decreasing, we can draw the conclusion about the drift and jump size of the model to guarantee the convex preserving property for the model defined in section 3. That is the coefficient

g

α

need not to be too convex and the jump size function is either nonnegative and convex or nonpositive and concave.

5 regularity for the value function and the parameters

Definition 5.1

z A function

f

on a metric space

E

⊂ ×

[0, ]

T

is locally Hölder(

α

) (denoted by )for

( )

C

_α

E

0

< <

α

1

if ,

|

( )

( ) |

sup

( , )

p q K

f p

f q

d p q

α ∈

−

_{< ∞}

for each compact

K

⊂

E

z A function

f

is at most polynomial growth on

E

⊂ ×

[0, ]

T

(denoted by

C

_pol

( )

E

):

0, 0

( )

{

( ) :|

( , ) |

(1 | | )

P pol C P

C

E

f

C E

f x t

C

x

> >

=

∪

∈

≤

+

for

( , )

x t

∈

E

}

Where

C E

( )

denotes all continuous functions on

E

.

z

C

p q,

( )

E

denotes all functions

f

in

C E

( )

such that all derivatives

D D f

_xk _tl with the condition

| | 2

k

+ ≤

l

p

and

0

≤ ≤

l

q

exist in the interior of

E

and with continuous extension to

E

.

z

C

_αp q,

( )

E

defines the set of all functions

f

∈

C

p q,

( )

E

such that

D D f

_xk _tl

∈

C

_α

( )

E

with

and .

| | 2

k

+ ≤ p

l

0

≤ ≤

l

q

A constant

D

>

1

exists such that:

a.1

|

β

( , ) |

x t

≤

D

(1

+

x

+

)

satisfies

β

∈

C

_α2,1

(

×

[0, ]

T

)

with

)

|

β

_t

( , ) |

x t

≤

D x

(| | 1

+

and

|

β

_xx

( , ) |

x t

≤

D x

(| | 1)

+

−1

a.2

| ( , ) |

α

x t

≤

D

(1 | |)

+

x

together with assumption

α

∈

C

_α2,1

(

×

[0, ]

T

)

and restrictions:

)

|

α

_t

( , ) |

x t

≤

D x

(| | 1

+

and

|

α

_xx

( , ) |

x t

≤

D x

(| | 1)

+

−1;

a.3

| ( , , ) |

φ

x t z

≤

D

(1 | |)

+

x

,

|

φ

_t

( , ) |

x t

≤

D x

(| | 1)

+

and

|

φ

_xx

( , ) |

x t

≤

D x

(| | 1)

+

−1 with

2,1

(

[0, ]

C

_α

T

φ

∈

×

)

a.4

| ( )

g x

−

g y

( ) |

≤

D x

|

−

y

|

and

g

∈

C

3_pol

( )

a.5

| |

x

≤ −

D

1

and locally Hölder

(

α

)

in

[0, ]

T

;

a.6

λ

∈

C

_α

([0, ])

T

(condition to use theorem A.11 in Janson and Tysk [6])

We note that assumptions a.1-a.3 and a.5 satisfy conditions (B2), (B22,1), (C1) defined in Janson and Tysk [6].

6 Sufficient conditions for preservation of convexity

In section 3, we discuss the LCP conditions for the jump-diffusion term structure model and derive the equation (4.3) as a conclusion. Moreover, following theorem 4.2 in Ekström and Tysk [1], equation (4.3) is sufficient conditions for LCP of the model as well. Then it is straightforward to examine if the LCP of the current model is sufficient for the spatially convexity at all times

.

[0, ]

t

∈

T

LEMMA 6.1 Under appropriate assumptions in section 5, The value function

U

is in . Moreover, it exists a constant and such that

4,1

( , )

(

[0, ])

U x t

⊆

C

_α

×

T

m

>

0

K

>

0

2

( , )

(

m

1)

U x t

≤

K x

+

+

for all

( , )

x t

∈ ×

[0, ]

T

.

Proof: Firstly, as well as PIDE defined in Pham [5](see equation (5.5) in Pham [5]), the corresponding PIDE in our jump term structure model is:

2

1

( ( ), )

(

)

( ( ), )

(

)

2

x t xx t

v

_τ

=

α

X t t v

X

+

β

X t t v

X

−

xv

1

υ

1

( ( ), )

( ( ), )

( ( ), , ) (

)

0

( ( )

( ( ), , ))

( ( )) (

)

v X t

φ

X t t z

v X t

dz

+

_∫

+

−

(6.1) where 0

X t t

X t t

X t t z

dz

α

=

α

−

∫

φ

υ

.

Under the assumptions in section 5, it is obvious that conditions (2.2)-(2.5) in Pham [5] are fulfilled. Then it is from theorem 3.1 in Pham [5] for Cauchy problem that there is existence of a viscosity solution for PIDE equation (6.1)

In order to use convexity property of solutions to parabolic equations, the equation (6.1) can be written in the way similar to Equation (A.6) in Janson and Tysk [6]:

2

1

( ( ), )

(

)

( ( ), )

(

)

( , )

2

( , 0)

( )

x t xx t

v

X t t v X

X t t v

X

xv

H x

v x

g x

τ

α

β

⎧ −

−

+

=

⎪

⎨

⎪

₌

⎩

t

(6.2) and we have . 1 0

( , )

( ( )

( ( ), , ))

( ( )) (

)

H x t

=

∫

v X t

+

φ

X t t z

−

v X t

υ

dz

In addition, proposition 3.3 in Pham [5] remains applicable for our case:

1/2

| ( , )

v t x

−

v s y

( , ) |

≤

C

(| (1 | |) |

+

x

t

−

s

|

+

|

x

−

y

|)

(6.3) It follows the assumptions a.3, a.4 and (6.3) that

H x t

( , )

and satisfy conditions (A.7) and (A.8) in Janson and Tysk [6]. So theorem A.7 in Janson and Tysk [6] can be applied in our function , if this is also the classical solution to (6.2).

( )

g x

( , )

v t x

v t x

( , )

Due to the theorem A.20 and A.18 in Janson and Tysk [6], the equation (6.2) has a unique solution

v

⊆

C

2,1_pol

(

×

[0, ])

T

∩

C

_α2,1

(

×

[0, ]

T

)

. Then we further have

H x t

( , )

belonging to . Now it is able to apply theorem A.11 in Janson and Tysk [6] to derive the result 2,0

(

[0, ]

C

_α

×

T

)

4,1

(

[0, ]

v

⊆

C

_α

×

T

)

. .

Moreover, the classic solution

v

is a viscosity solution for (6.2) as well, together with the uniqueness theorem in Pham [5], we obtain

v

≡

v

.

In conclusion, we have

U x t

( , )

⊆

C

_α4,1

(

×

[0, ])

T

and

U x t

( , )

≤

K x

(

m+2

+

1)

where

C

is a constant depending on , and . As a result the second derivative of satisfies the condition

T

m

D

U x t

( , )

( , )

m

xx

U

x t

≤

K x

. Assumption 6.1

f

is smooth, concave, and there exists a constant

K

'

>

0

such that

( )

x

f x

const

⎧

= ⎨

⎩

'

'

f x

K

if x

K

≤

>

(6.4) Proposition 6.1 Assume that

α

_xx

− ≤

2

0

(in the sense of distribution) at all points

( , )

x t

∈

[0, ]

T

×

and

φ

_xx

( , , ) ( , , )

x t z

φ

x t z

≥

0

for all points

( , , )

x t z

∈ ×

[0, ] [0,1]

T

×

.Also let assumptions of Lemma 6.1 hold. Then, if the pay-off function

g

is convex, the corresponding

value function

U x t

( , )

is globally convex at all points

t

∈

[0, ]

T

.

Note: we define

V x

( , )

τ

solve the corresponding initial value problem:

( , 0)

( )

V

LV

fV

V x

g x

τ

=

−

⎧

⎨

₌

⎩

(6.5) Where

L

has already been defined in section 3 (see (3.3)).

Proof: We firstly study the global convexity property of function

V

, in the conditions:

2

0

( , , ) ( , , )

0

xx x xx

f

x t z

x t z

α

φ

φ

−

≤

⎧

⎨

_≥

⎩

(6.6) Similar to the analytical way in Ekström and Tysk [3], it is intuitive to consider the function:

(( 1) 1)

2 ( )

( , ) :

( , )

M

(

m

1)

f x e D

V

ε

x

τ

=

V x

τ

+

ε

e

τ

X

+

+

e

− λ+ + τ (6.7) Here, we assume

ε

>

0

and

m

>

0

is an even number so that

(

X

m+2

+

1)

e

−f x e( ) ((λ+1)D+1)τ has a strictly positive second derivative at all points

( , )

x

τ

∈ ×

[0, ]

T

. Meanwhile, the operator of

( , )

V x

τ

satisfies

M

f

:

= −

L

f

, where

L

is defined in (3.3).

For convenience, we denote

h x

( )

=

X

m+2

+

1

and

p

=

e

−f x e( ) ((λ+1)D+1)τ then we have:

2 2

(

f

)

M

(

m x

V

M V

e

M

x

X

p

ε ε τ τ

ε

⎡

+

⎤

∂ ∂

−

=

∂

_⎣

2

+

_⎦

1)

1)

fhp

τ

]

)

−

2 2 (( 1)

[

(

)

(

)

((

1)

1)

M D x x x

e

τ

hp

hp

fhp

D

e

λ

ε

α

β

λ

+ +

−

∂

∂

+ ∂

−

+

+

+

1 0

h x

(

) (

p x

)

h x p

( )

x

(

hp dz

)

λ

φ

φ

φ

+

_∫

+

+

−

− ∂

1 2

(

M

e

τ

MI

I

ε

=

(6.8) For large positive

x

, we have

f

constant and therefore

e

−f x( ) is bounded. As a result, we obtain

I

₁

~

X

m, and

I

₂ grows at most like

X

m(with

φ

under the assumption in section 5).

For large negative

x

, we have

f

=

x

. Then

I

₁

~

X

m+2

p

)

, whereas 2 (( 1) 1) ( 2) 2

~

(

((

1)

1)

D D xxxx x

I

β

hp

+ ∂ −

α

e

λ+ + τ

hp

−

xhp

+

λ

+

D

+

e

+ τ

xhp

1 1

dz

2 (( 1) 1) 2 0

(

)

0

( (

) (

)

( ) )

)

D x

e

hp dz

x

h x

p x

h x p

λ τ

λ

φ

+ +

λ

φ

φ

+

∫

∂

+

∫

∂

+

+

−

Using assumptions for

β

,

α

and

φ

, we forward derive: (( 1) 1) (( 1) 1) 2

~

(

1 ((

1)

1)

)

D D xxxx xx

I

β

hp

+ −

De

λ+ + τ

− +

λ

+

D

+

e

λ+ + τ

xhp

1

)

D

_≥

(( 1) 1) 2 0

(

)(1

)

( )

D xx x xx xx

De

λ τ

xhp

h x

p

h x p dz

λ

+ +

λ

φ

φ

−

+

_∫

+

+

−

We notice that: (( 1) 1) (( 1) 1) (( 1) 1)

1 ((

1)

1)

0

D D

De

λ+ + τ

λ

D

e

λ+ + τ

λ

De

λ+ + τ

−

− +

+

+

−

Since both

β

and

φ

_xis bounded for negative

x

, We have

2 2

~

m

I

X

+

p

Finally,

I

₁grows at least as fast as

I

₂while

| |

x

tends to infinite. Then,

M

can be chosen large enough so that positive term

MI

₁dominates

I

₂everywhere. In another way, M is chosen to guarantee that 1 2

0

MI

−

I

>

(6.9) Then we define:

: {( , ) :

_xx

( , )

0}

E

=

x

τ

V

ε

x

τ

<

.

Our mission is to proof that

E

is empty. It Follows the similar procedure in Ekström and Tysk [3] to have

E

satisfy that

E

⊆ −

[

ρ ρ

, ]

×

for some positive

ρ

(using Lemma 6.1 and growth rate of

hp

which we have done above). Then we say

E

is bounded, so

E

is compact. Since it exists an infimum:

0

: inf{

0 : ( , )

x t

E

τ

=

τ

≥

∈

for some

x

∈

}

Since value function and , we have

which means

( , 0)

( )

V x

=

g x

g

_xx

≥

0

V

_xxε

( , 0)

x

≥ ∂

ε

_x2

(

he

−f x( )

)

>

0

0

0

τ

>

. Due to the continuity of

V

_xx, we have

V

_xxε

(

x

₀

,

τ

₀

)

=

0

for some

x

₀. Consequently, for

τ τ

<

₀,

V

_xxε must satisfy

V

_xxε

(

x

₀

, )

τ

≥

0

. The second derivative of

V

ε is decreasing at

τ

₀: 0 0

(

,

)

0

xx

V

ε

x

τ

τ

∂

≤

(6.10) According to the sufficient conditions for LCP derived in section 4, assumptions in (6.6) is sufficient to yield the LCP of

V

ε

( , )

x

τ

with operator

M

f

:

= −

L

f

.

definition of LCP gives: 2 0 0

(

f

(

,

))

x

M V

x

ε

_τ

0

∂

≥

(6.11) It is straightforward to calculate the formula in (6.8) by using outcomes (6.10) and (6.11):

2 2 0 0 0 0 0 0 0 0

(

(

,

)

f

(

,

))

(

,

)

(

f

(

,

))

0

x

V

x

M V

x

V

xx

x

x

M V

x

ε ε ε ε τ

τ

τ

τ

τ

τ

∂ ∂

−

= ∂

− ∂

≤

(6.12)

which contradicts the result obtained in (6.9). Hence, we claim that the set

E

is empty. Finally, the value function

V

ε

( , )

x

τ

is globally convex under assumption (6.6), and let

ε

tends to zero to obtain the globally convexity of

V x

( , )

τ

. Furthermore, by monotone convergence, the convexity of

U x

( , )

τ

is obtained.

7. Monotonicity in the parameters

As discussed in the beginning, monotonicity of the model parameters is one of the main reasons to study convexity property. In this section, we study the monotonicity property for the shrift coefficient

α

( , )

x t

, diffusion coefficient

β

( , )

x t

, jump intensity

λ

( )

t

and jump size

φ

( , , )

x t z

. Following consequences in Ekström and Tysk [2] and Ekström and Tysk [3], for any convex and decreasing pay-off function satisfying assumption in section 5, the value function is probably decreasing in shift and increasing in volatility, jump intensity and jump size, if the model is globally convex for sure.

( )

g x

Theorem 7.1 Assume that

α

( , )

x t

≤

α

( , )

x t

,

β

( , )

x t

≥

β

( , )

x t

and

λ

( )

t

≥

λ

( )

t

. The pay-off function is convex and decreasing under the assumption in section 5. In addition, the jump size function

( )

g x

( , , )

x t z

φ

fulfills:

( , , )

1

( , , )

x t z

x t z

φ

φ

≥

for all points

( , , )

x t z

∈ ×

[0, ] [0,1]

T

×

and

φ

( , , )

x t z

≠

0

. If either

α

( , )

x t

and

φ

( , , )

x t z

or

( , )

x t

α

and

φ

( , , )

x t z

satisfy the conditions in proposition 6.1, then it follows

U x t

( , )

≥

U x t

( , )

. Proof: The formal procedure is simply inspired by schedule in proposition 6.1 integrating methods used in Ekström and Tysk [2] and Ekström and Tysk [3].

As in proposition 6.1, we define

V x

( , )

τ

as the form of (6.5) with

f

satisfying (6.4). Then it is first to study the monotonicity property with respect to value function

V x

( , )

τ

. We then consider the function

V

ε

( , )

x

τ

as defined in (6.7).

Then, assume

: {( , ) :

( , )

( , )

0}

E

=

x

τ

V

ε

x

τ

−

V x

τ

<

is not empty. From lemma 6.1 and proposition 6.1, it is known that

V x

( , )

τ

and

V

ε

( , )

x

τ

grows like

x

m+2. Then the set

E

is bounded,

E

is compact. Hence, there is an infimum:

0

: inf{

0 : ( , )

x t

E

τ

=

τ

≥

∈

for some

x

∈

}

Since

E

is compact and

V

ε

( , )

x

τ

−

V x

( , )

τ

is continuous, it exists a point

(

x

₀

,

τ

₀

)

which satisfies

V

ε

(

x

₀

,

τ

₀

)

−

V x

(

₀

,

τ

₀

)

=

0

. By the assumption on

g x

( )

, we have

τ

₀

>

0

( here the value function

V

ε

(

x

₀

, 0)

>

E

_{x t,}

[ (

g X

_T

)]

≥

E

_{x t,}

[ (

g X

_T

)]

=

V x

(

₀

, 0)

). So for

0

< <

τ τ

₀, we have

V

ε

(

x

₀

, )

τ

−

V x

(

₀

, )

τ

≥

0

, and that means:

0 0 0 0

(

V

ε

(

x

,

)

V x

(

,

))

τ

τ

τ

∂

−

≤

0

(7.1) On the other hand, equation (7.1) can be calculated directed by definition (6.5). It is first to define corresponding operator for

V

εand

V

:

:

f

M

= −

L

f

and

M

f

:

= −

L

f

where

L

is defined previously, and

(

)

1 2 0

1

(

)

( )

2

x xx x

LV

=

α

V

+

β

V

+

λ

∫

V x

+

φ

−

V x

−

φ

V

d

z

. Then, the value of left hand side in (7.1) takes:

0 0 0 0 1 2

(

V

ε

(

x

,

)

V x

(

,

))

M V

f ε

M V

f

e

M

(

MI

I

τ

τ

τ

ε

∂

−

=

−

+

τ

−

)

(7.2)

Here, by the same reasoning procedure in section 6, we can choose constant

M

large so that

(

MI

₁

−

I

₂

)

> 0

. Since, at point

(

x

₀

,

τ

₀

)

, the equation (7.2) satisfies:

(

V

ε

V

)

M V

f ε

M V

f τ

∂

−

>

−

2 2 1 1 2 2

(

α

V

_xε

α

V

_x

) (

β

V

_xxε

β

V

_xx

)

f V

(

V )

ε

=

−

+

−

+

−

1 0 0 0 0 0 0 0 0

[ ( )(

V

(

x

,

)

V

(

x

,

)

V

x

(

x

,

))

ε ε ε

λ τ

φ τ

τ

φ

+

_∫

+

−

−

τ

0 0 0 0 0 0 0

(

)( (

V x

,

)

V x

(

,

)

V x

_x

(

,

))]

dz

λ τ

φ τ

τ

φ

τ

−

+

−

−

Since the function

V

ε

( ,

x

τ

₀

)

−

V x

( ,

τ

₀

)

obtain its minimum zero at

x

=

x

₀ . At point

0 0

(

x

,

τ

)

we have

V

ε

=

V

,

V

_xε

=

V

_x

≤

0

and

V

_xxε

≥

V

_xx. From the assumption in the beginning, it is true that

V

_xxε

≥

V

_xx

≥

0

or

V

_xxε

≥ ≥

0

V

_xx holds. By Ekström and Tysk [2] and Ekström and Tysk [3], it is easy to proof:

0 0 0 0

(

V

ε

(

x

,

)

V x

(

,

))

τ

τ

τ

0

∂

−

>

(7.3) This contradicts inequality (7.1). Hence, the corresponding set

E

is empty and

( , )

( , )

0

V

ε

x

τ

−

V x

τ

≥

holds. Let

ε

→

0

together with

f

→

x

(by monotone convergence), it is also true that

U x

( , )

τ

≥

U x

( , )

τ

.

References

[1] E. Ekström, J. Tysk, Convexity preserving jump-diffusion models for option pricing, J. Math.

Anal. 330 (2007) 715-728.

[2] E. Ekström, J. Tysk, Properties of option prices in models with jumps, Mathematical Finance, Vol. 17, No. 3 (July 2007), 381-397.

[3] E. Ekström, J. Tysk, Convexity theory for the term structure equation, Finance Stoch (2008) 12: 117-147.

[4] R. Cont, P. Tankov (2004), Financial modeling with Jump Processes. Boca Raton, FL: Chapman & Hall.

[5] H. Pham (1998), Optimal Stopping of Controlled Jump Diffusion Processes: A Viscosity Solution Approach. J. Math. Systems Estim. Control 8, 1-27.

[6] S. Janson, J. Tysk, Preservation of convexity of solutions to parabolic equations. J. Diff. Equ. 206, 182-226 (2004).